1. Field of the Invention
This invention relates to seismic exploration and more particularly, to an adaptive method for cancelling nonstationary sinusoidal noise from seismic data.
2. Description of the Prior Art
In seismic exploration, it is common practice to deploy a large array of geophones on the surface of the earth and to record the vibrations of the earth at each geophone location to obtain a collection of seismic traces. The traces are sampled and recorded for further processing. When the vibrations so recorded are caused by a seismic source activated at a known time and location, the recorded data can be processed by a computer in known ways to produce an image of the subsurface. The image thus produced is commonly interpreted by geophysicists to detect the possible presence of valuable hydrocarbons.
Seismograms are commonly recorded as digital samples representing the amplitude of a received seismic signal as a function of time. Since seismograms are usually obtained along a line of exploration on the surface of the earth, the digital samples can be formed into x-t arrays with each sample in the array representing the amplitude of the seismic signal as a function of horizontal distance and time. When such arrays are visually reproduced, by plotting or the like, seismic sections are produced. A seismic section depicts the subsurface layering of a section of the earth. It is the principal tool which the geophysicist studies to determine the nature of the earth's subsurface formations. Before an array of seismic samples or traces can be converted into a seismic section for interpretation by geophysicists, the array must be extensively processed to remove noise and to make reflection events discernible.
In the course of seismic exploration, a significant amount of noise is introduced into the seismic data by the influence of man. Such introduced noise, which may include noise from power lines, electrical machinery or other sources, is generally referred to as "cultural noise". The usual method of estimating a signal corrupted by additive noise is to pass it through a filter that tends to suppress the noise while leaving the signal relatively unchanged. Filters for this purpose may be either fixed or adaptive. The design of fixed filters is based on prior knowledge of both the signal and the noise. The traditional method of removing narrow band noise contamination utilizing a fixed filter design is to pass the seismic data through a notched filter. However, the use of such prior art notched filters often gives poor results. First, the notch might not be centered exactly at the frequency of the noise contamination. Second, the notch removes a considerable amount of the valid data along with the noise. Third, the contamination from the noise often extends to parts of the amplitude spectrum outside of the notch with the result that there is noise remaining in the data after the notch filter has been applied. Finally, in some cases, the notch filter can induce phase distortion in the data. Unlike fixed filters, adaptive filters, however, have the ability to adjust their own parameters automatically and their design requires little or no prior knowledge of signal or noise characteristics. See, for example, U.S. Pat. No. 3,889,108 issued to Cantrell.
Noise cancelling is a variation of optimal filtering where a reference input is filtered and subtracted from a primary input containing both signal and noise to attenuate or eliminate primary noise by cancellation Bernard Widrow et al, "Adaptive Noise Cancelling: Principles and Applications", Proceedings of the IEEE, Vol. 63, No. 12, December 1975, describe a method of using a "primary" input containing the corrupted signal and a "reference" input containing noise correlated in some way with the primary noise where the reference input is adaptively filtered and subtracted from the primary input to obtain the signal estimate.
The adaptive noise cancellation apparatus described by Widrow et al may be seen by reference to FIG. 1. The primary input is assumed to be any kind of signal--stochastic, deterministic, periodic, transient, etc.--or any combination of signals. The reference input is assumed to be a pure cosine wave C cos(.OMEGA..sub.0 t+.PHI.). The primary and reference inputs are sampled at a sampling frequency of 2.pi./T radians/second. The reference input is sampled directly, giving x.sub.1k, and after undergoing a 90 degree phase shift, giving x.sub.2k. The sampled reference inputs are provided to the basic element of the Widrow adaptive noise cancelling system--an LMS adaptive filter, the principal component of which is an adaptive linear combiner for weighting and summing a set of input signals to form an output signal. The LMS adaptive algorithm is designed to adjust the weights of the adaptive linear combiner to minimize mean-square error. Operating as an implementation of the method of steepest descent, the LMS algorithm adaptively supplies estimations of the amplitudes of reference inputs x.sub.1k and x.sub.2k. Similar apparatus may be seen by reference to Bernard Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice-Hall, Inc., 1985, pgs. 99-116 and 316-337; and J. R. Glover, Adaptive Noise Cancelling of Sinusoidal Interference, Stanford University dissertation published by University Microfilms International, 1986, pgs. 30-34 and 64-72.
Widrow et al sought to apply noise cancellation to remove nonstationary sinusoidal noise contaminating a primary signal. In Widrow et al, however, the contaminating sinusoid wa also available as a separate signal. To accomplish the objective using two external signals, it was necessary to adjust only the amplitude and phase of the sinusoidal signal to match that found in the primary signal. This objective was readily satisfied by adjusting the two amplitudes of the quadrature components of the sinusoidal signal. Widrow et al are silent, however, as to any teaching or suggestion of permitting the frequency or phase of the quadrature components of the sinusoidal signal to change
The problem which arose during attempts to apply the teachings of Widrow to seismic applications was that there was no recording available for the contaminating sinusoid. Attempts were made to employ a synthetic sinusoidal signal to approximate the contaminating sinusoid. It was felt that this might be successful if the contaminating frequency could be estimated from a spectral analysis of the seismic recording. This implementation worked, however, only if the frequency did not change appreciably--e.g. the two frequencies never deviated by more than 1 or 2 per cent over the duration of the seismic recording. Since this condition cannot always be satisfied, the Widrow technique has proven unsatisfactory for many seismic applications.